Phenylated polyimides: Diels-Alder reaction of biscyclopentadienones with dimaleimides

A series of phenylated polydihydrophthalimides has been synthesized by the Diels-Alder reactions of 3,3′-(oxydi-p-phenylene)bis(2,4,5-triphenylcyclopentadienone) and 3,3′-(p-phenylene)bis(2,4,5-triphenylcyclopentadienone) with N,N′-o-, -m-, and -p-phenylenedimaleimide. The polydihydrophthalimides were soluble in dimethylformamide (DMF) and had intrinsic viscosities that ranged from 0.33 to 1.01, the polymers were dehydrogenated thermally and chemically to afford the corresponding phenylated polyphthalimides. The totally aromatic polyimides were also soluble in DMF but had intrinsic viscosities only as high as 0.41. The thermogravimetric analyses of the polyphthalimides showed breaks near 530°C in air and in nitrogen atmospheres.     

Frobenius distributions of elliptic curves in short intervals on average

The Ramanujan Journal - For an elliptic curve $$E/{\mathbb {Q}}$$ , let $$a_p$$ denote the trace of its Frobenius endomorphism over $${\mathbb {F}}_p$$ , where p is a prime of good reduction for E....